Function THE-DYAD

• Defined in theory: Tensor-quantities
• Source code: tensor-quantities.lisp

Slots on this function:

Documentation:

Constructor function for second order tensors. Requires a matrix and an orthnormal-basis as arguments. The constructed dyad is equal to the sum of products of matrix elements with the corresponding basis vectors.

Arity: 3

Axioms:

(Nth-Domain The-Dyad 3 Dyad)

(Nth-Domain The-Dyad 2 Orthonormal-Basis)

(Nth-Domain The-Dyad 1 Square-Matrix)

(Nth-Domain The-Dyad 1 Matrix-Quantity)

(<=> (The-Dyad ?M ?B ?T)
     (And (Matrix-Quantity ?M)
          (Square-Matrix ?M)
          (Orthonormal-Basis ?B)
          (= (Quantity.Dimension ?T) (Quantity.Dimension ?M))
          (= (Size ?M) (Basis.Dimension ?B))
          (= (Spatial.Dimension ?T) (Size ?M))
          (Dyad ?T)
          (= ?T
             (Summation (Lambda (?I)
                                (Summation (Lambda (?J)
                                                   (* (Basis.Vec ?B
                                                                 ?I)
                                                      (* (Value ?M
                                                                ?I
                                                                ?J)
                                                         (Basis.Vec ?B
                                                                    ?J))))
                                           1
                                           (Spatial.Dimension ?T)))
                        1
                        (Spatial.Dimension ?T)))))

Notes:

• Example: (= (the-dyad (matrix-of-columns (matrix-of-rows (* 3 feet) (* 1 yard)) (matrix-of-rows (* 24 inches) (72 inches))) 2d-frame-n) (* (the-dyad (matrix-of-rows (matrix-of-columns 3 2) (matrix-of-columns 3 6)) 2d-frame-n) feet))